A276367 - OEIS

%I #10 Dec 03 2017 04:55:45

%S 1,2,8,46,322,2564,22482,213358,2170856,23563266,272229894,3345403228,

%T 43736868406,608546129090,9012054592672,141977836757366,

%U 2376612322575818,42191783298374292,792519258202255050,15709695283993859430,327743321824492243272,7177487348025844367658,164595689482728908058190,3943617273778939651118764,98517855256524601996722238,2561403841975017528679295466,69192589389178960801205055872

%N G.f.: A(x) = Sum_{n>=1} Product_{k=0..n-1} (x + k*A(x)).

%H Vaclav Kotesovec, <a href="/A276367/b276367.txt">Table of n, a(n) for n = 1..125</a> (terms 1..100 from Paul D. Hanna)

%F G.f.: A(x) = Sum_{n>=1} Sum_{k=1..n} |S1(n,k)| * x^k * A(x)^(n-k), where |S1(n,k)| = A000254(n,k) form the unsigned Stirling numbers of first kind.

%e G.f.: A(x) = x + 2*x^2 + 8*x^3 + 46*x^4 + 322*x^5 + 2564*x^6 + 22482*x^7 + 213358*x^8 + 2170856*x^9 + 23563266*x^10 + 272229894*x^11 + 3345403228*x^12 + 43736868406*x^13 + 608546129090*x^14 + 9012054592672*x^15 + 141977836757366*x^16 +...

%e such that

%e A(x) = x + x*(x + A(x)) + x*(x + A(x))*(x + 2*A(x)) + x*(x + A(x))*(x + 2*A(x))*(x + 3*A(x)) + x*(x + A(x))*(x + 2*A(x))*(x + 3*A(x))*(x + 4*A(x)) + x*(x + A(x))*(x + 2*A(x))*(x + 3*A(x))*(x + 4*A(x))*(x + 5*A(x)) +...

%o (PARI) {a(n) = my(A=x); for(i=1,n, A = sum(m=1,30, prod(k=0,m-1, x + k*A +x*O(x^n)))); polcoeff(A,n)}

%o for(n=1,30,print1(a(n),", "))

%o (PARI) {a(n) = my(A=x); for(i=0,n, A = sum(m=1,n, sum(k=1,m, abs( stirling(m,k,1) )*x^k*(A + x*O(x^n))^(m-k) ) ) ); polcoeff(A,n)}

%o for(n=1,30,print1(a(n),", "))

%K nonn

%O 1,2

%A _Paul D. Hanna_, Sep 02 2016